# Relationship between magnetic resonance linewidth and spin relaxation

First of all, what is the mathematical relationship between measured linewidth (usually in units of magnetic field) and spin relaxation time? I see papers talk about spin relaxation times in terms of linewidths but I have no idea how to correlate the size of the linewidth to an actual time.

Second, is the measured linewidth in NMR or ESR experiments related to $T_2^{\ast}$ only. If you want just $T_2$, you would have to do spin-echo, right? If that is the case, how is $T_1$ determined?

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Is linewidth the image-resolution? – kame Feb 26 '11 at 22:05
Very naively speaking, linewidth ~ $\Delta E \propto 1/\Delta T$, where $\Delta T$ is the spin relaxation time. – user346 Feb 27 '11 at 3:18

The theory is rather generic Fourier transform: if you have a perfectly non-decaying oscillation $e^{i\omega_0 t}$ (with real $k$) then the transform gives a perfectly sharp spectrum as a delta function $\delta(\omega - \omega_0)$. But if the excitation decays $e^{i\omega_0 t} e^{-k t}$ then we get $\delta(\omega - \omega_0 - i k)$, which has a real part that is broadened: $1/\left({(\omega-\omega_0)^2 + k^2}\right)$.
Now, usually this manifests as some sort of sweep time, as experimentally you would drive the system at some changing frequency $\omega$ and observe the driven amplitude.